2-Rainbow Domination Number of the Subdivision of Graphs

Document Type : Original paper


Department of Mathematics, ‎Faculty of Science, Imam Khomeini International University


Let $G$ be a simple graph and $f : V (G) \rightarrow P(\{1,2\})$ be a function where for each vertex $v \in V (G)$ with $f(v)= \emptyset$ we have $\bigcup_{u \in N_{G}(v)} f(u) = \{1,2\}.$ Then $f$ is a $2$-rainbow dominating function (a $2RDF$) of $G.$ The  weight of $f$ is $\omega(f)=\sum_{v \in V(G)} |f(v)|.$ The minimum weight among all of $2-$rainbow dominating functions is $2-$rainbow domination number  and is denoted by $\gamma_{r2}(G)$. In this paper,  we provide some bounds for the $2-$rainbow domination number of the subdivision graph $S(G)$ of  a graph $G$. Also, among some other interesting results, we determine the exact value of $\gamma_{r2}(S(G))$ when $G$ is a tree, a bipartite graph, $K_{r,s}$, $K_{n_1,n_2,\dots,n_k}$ and $K_n$.


Main Subjects

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